Binomial Probability Calculator
Calculate the probability of exactly k successes in n independent Bernoulli trials with success probability p.
P(X = k)
0.11718750
P(X = k) vs Number of Trials (n)
Formula
## How to Calculate Binomial Probability ### Formula **P(X = k) = C(n,k) * p^k * (1-p)^(n-k)** The binomial distribution models the number of successes in n independent trials, each with the same probability p. C(n,k) counts the number of ways to arrange k successes among n trials. The mean is np and the variance is np(1-p).
Esempio Risolto
Flip a fair coin 10 times. What is the probability of exactly 3 heads?
- 01C(10,3) = 10! / (3! * 7!) = 120
- 02p^k = 0.5^3 = 0.125
- 03(1-p)^(n-k) = 0.5^7 = 0.0078125
- 04P(X=3) = 120 * 0.125 * 0.0078125 = 0.1171875
- 05Expected value = 10 * 0.5 = 5
Domande Frequenti
What are the requirements for a binomial distribution?
Fixed number of trials (n), each trial has exactly two outcomes (success/failure), trials are independent, and the probability of success (p) is constant across all trials.
How does the binomial relate to the normal distribution?
When n is large, the binomial distribution is approximately normal with mean np and standard deviation sqrt(np(1-p)). A common rule of thumb is np >= 5 and n(1-p) >= 5.
What if I need P(X >= k) instead of P(X = k)?
You would sum P(X = k) + P(X = k+1) + ... + P(X = n). Alternatively, use the complement: P(X >= k) = 1 - P(X < k) = 1 - sum of P(X = 0) through P(X = k-1).
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